The curse of dimensionality gives rise to prohibitive computational requirements that render infeasible the exact solution of large-scale stochastic control problems. We study an efficient method based on linear programming for approximating solutions to such problems. The approach “fits ” a linear combination of pre-selected basis functions to the dynamic programming cost-to-go function. We develop error bounds that offer performance guarantees and also guide the selection of both basis functions and “state-relevance weights ” that influence quality of the approximation. Experimental results in the domain of queueing network control provide empirical support for the methodology. (Dynamic programming/optimal control: approximations/large-scale problems. Queues, algorithms: control of queueing networks.)

We present a kernel-based approach to reinforcement learning that overcomes the stability problems of temporal-difference learning in continuous state-spaces. First, our algorithm converges to a unique solution of an approximate Bellman's equation regardless of its initialization values. Second, the method is consistent in the sense that the resulting policy converges asymptotically to the optimal policy. Parametric value function estimates such as neural networks do not possess this property. Our kernel-based approach also allows us to show that the limiting distribution of the value function estimate is a Gaussian process. This information is useful in studying the bias-variance tradeo in reinforcement learning. We find that all reinforcement learning approaches to estimating the value function, parametric or non-parametric, are subject to a bias. This bias is typically larger in reinforcement learning than in a comparable regression problem.

Abstract We develop a new method for pricing American options. The main practical contribution of this paper is a general algorithm for constructing upper and lower bounds on the true price of the option using any approximation to the option price. We show that our bounds are tight, so that if the initial approximation is close to the true price of the option, the bounds are also guaranteed to be close. We also explicitly characterize the worst-case performance of the pricing bounds. The computation of the lower bound is straightforward and relies on simulating the suboptimal exercise strategy implied by the approximate option price. The upper bound is also computed using Monte Carlo simulation. This is made feasible by the representation of the American option price as a solution of a properly defined dual minimization problem, which is the main theoretical result of this paper. Our algorithm proves to be accurate on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. These numerical results suggest that our pricing method can be successfully applied to problems of practical interest.

The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Dual pricing of multi-exercise options under volume constraints

SUMMARY In the 20 years following the publication of the ARCH model, there has been a vast quantity of research uncovering the properties of competing volatility models. Wide-ranging applications to financial data have discovered important stylized facts and illustrated both the strengths and weaknesses of the models. There are now many surveys of this literature. This paper looks forward to identify promising areas of new research. The paper lists five new frontiers. It briefly discusses three-high-frequency volatility models, large-scale multivariate ARCH models, and derivatives pricing models. Two further frontiers are examined in more detail-application of ARCH models to the broad class of non-negative processes, and use of Least Squares Monte Carlo to examine non-linear properties of any model that can be simulated. Using this methodology, the paper analyses more general types of ARCH models, stochastic volatility models, long-memory models and breaking volatility models. The volatility of volatility is defined, estimated and compared with option-implied volatilities.

Abstract not found

Analytical tractability is one of the challenges faced by many alternative models that try to generalize the Black-Scholes option pricing model to incorporate more empirical features. The aim of this paper is to extend the analytical tractability of the Black-Scholes model to alternative models with jumps. We demonstrate a double exponential jump diffusion model can lead to an analytic approximation for Þnite horizon American options (by extending the Barone-Adesi and Whaley method) and analytical solutions for popular path-dependent options (such as lookback, barrier, and perpetual American options). Numerical examples indicate that the formulae are easy to be implemented and accurate.

Abstract not found

Although traded as distinct products, caps and swaptions are linked by noarbitrage relations through the correlation structure of interest rates. Using a string market model, we solve for the correlation matrix implied by swaptions and examine the relative valuation of caps and swaptions. We find that swaption prices are generated by four factors and that implied correlations are lower than historical correlations. Long-dated swaptions appear mispriced and there were major pricing distortions during the 1998 hedge-fund crisis. Cap prices periodically deviate significantly from the no-arbitrage values implied by the swaptions market. THE GROWTH IN INTEREST-RATE SWAPS during the past decade has led to the creation and rapid expansion of markets for two important types of swap-related derivatives: interest-rate caps and swaptions. These over-the-counter derivatives are widely used by many firms to manage their interest-rate risk exposure and collectively represent the largest class of fixed-income options in the financial markets. The International Swaps and Derivatives Association

We develop a structural bond valuation model to simultaneously capture liquidity and credit risk. Our model implies that renegotiation in financial distress is influenced by the illiquidity of the market for distressed debt. As default becomes more likely, the components of bond yield spreads attributable to illiquidity increase. When we consider finite maturity debt, we find decreasing and convex term structures of liquidity spreads. Using bond price data spanning 15 years, we find evidence of a positive correlation between the illiquidity and default componentsofyieldspreadsaswellassupportfordownward-slopingtermstructuresof liquidity spreads. Credit risk and liquidity risk have long been perceived as two of the main justifications for the existence of yield spreads above benchmark Treasury notes or bonds (see Fisher (1959)). Since Merton (1974), a rapidly growing body of literature has focused on credit risk. 1 However, while concern about market liquidity issues has become increasingly marked since the autumn of 1998, 2 liquidity remains a relatively unexplored topic, in particular, liquidity for defaultable securities. 3 This paper develops a structural bond pricing model with liquidity and credit risk. The purpose is to enhance our understanding of both the interaction between these two sources of risk and their relative contributions to the yield spreads on corporate bonds. Throughout the paper, we define liquidity as the ability to sell a security promptly and at a price close to its value in frictionless markets, that is, we think of an illiquid market as one in which a sizeable discount may have to be incurred to achieve immediacy. We model credit risk in a framework that allows for debt renegotiation as in Fan and Sundaresan (2000). Following François and Morellec (2004), we also introduce